MODTRAN4 is the U.S. Air Force (USAF) standard moderate (2 cm−1) or broader/coarser spectral resolution radiative transport (RT) model for wavelengths extending from the thermal InfraRed (IR) through the visible and into the ultraviolet (0.2 to 10,000.0 μm). [See: A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, J. H. Chetwynd, M. W. Matthew, E. P. Shettle and S. M. Adler-Golden, “MODTRAN4 User's Manual,” Air Force Research Laboratory Report, June 1999, and see: A. Berk, L. S. Bernstein, G. P. Anderson, P. K. Acharya, D. C. Robertson, J. H. Chetwynd and S. M. Adler-Golden, “MODTRAN Cloud and Multiple Scattering Upgrades with Application to AVIRIS,” Remote Sens. Environ. 65, pp. 367-375, 1998.]
The MODTRAN4 1 cm−1 statistical band model (from which 2 cm−1 spectral resolution results are obtained) provides a fast alternative (100-fold increase in speed) to the USAF more accurate line-by-line (LBL) radiative transport models, FASCODE and FASE. [See: S. A. Clough, F. X. Kneizys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abreu, and L. A. Hall, “FASCODE3 Spectral Simulation,” Proceedings of the International Radiation Symposium, Lenoble and Geleyn, Deepak Publishing, 1988, and see: H. E. Snell, J. -L. Moncet, G. P. Anderson, J. H. Chetwynd, S. Miller, and J. Wang, “FASCODE for the Environment (FASE)”, Proceedings of Atmospheric Propagation and Remote Sensing IV, SPIE, 2471, pp. 88-95, 1995.] FASCODE and FASE are both based on a ‘first principles’ physical equations, expanding the optical depth terms, based on spectroscopic constants and line shape, with very high accuracy. Comparisons between MODTRAN4 and FASE spectral transmittances and radiances show agreement to within a few percent or better in the thermal IR. MODTRAN4 includes flux and atmosphere-scattered solar calculations, essential components in analysis of near-IR, visible and ultraviolet spectral region data that are not readily generated by LBL models.
MODTRAN4 and its predecessors have been used extensively over the last quarter century in the design and analysis of broadband, multiband, and short-wave IR/Visible hyperspectral imaging sensors. However, conventional interferometers and many state-of-the-art sensors working in the long- and mid-wave IR operate at higher spectral resolution than MODTRAN4 provides.
Narrowing the band model spectral resolution changes the fundamental character of the band model. The half-width of molecular transitions near sea level average about 0.08 cm−1. As illustrated in FIG. 1(a), the 1.0 cm−1 band model calculates the absorption of atomic and molecular lines whose line center regions lie almost entirely within the spectral bin. At the finer spectral resolution, a much larger fraction of any atomic or molecular line falls outside of the spectral bin containing the line. Determination of the new band model has therefore required improved treatment of both line tail and line center absorption. Line tail absorption is modeled closer to line centers (as defined by a compilation of spectroscopic data), and the finite-bin single-line equivalent width used to calculate line center absorption is no longer simply a small perturbation of the total single line equivalent width.
The line center absorption within a spectral bin is generally defined as the in-band absorption from all atomic and molecular transitions centered in that bin, FIG. 2. LBL models calculate this in-band absorption by explicitly determining the spectral absorption of each line on a very fine spectral grid and then integrating the resulting spectrum. In a band model approach, the in-band absorption is approximated based on statistical assumptions regarding line positions and overlap. Temperature dependent band model parameters are computed from an atomic and molecular transition line atlas such as HITRAN. [See L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. -M. Flaud, A. Perrin, V. Dana, J.-. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, The HITRAN Atomic and molecular Spectroscopic Database and HAWKS (HITRAN Atmospheric Workstation): 1996 Edition, J. Quant. Spectrosc. Radiat. Transfer, 60, pp. 665-710 (1998)]. These parameters define an effective single-line for the interval, characterized by its absorption line strength and half-width parameters, and the effective number of lines, neff.
In MODTRAN4, the finite spectral bin single-line Voigt equivalent width Wsl is computed to determine the absorption of the effective average line. It is calculated as the difference between the total equivalent width, computed using the Rodgers-Williams formula [see: C. D. Rodgers and A. P. Williams, “Integrated absorption of a spectral line with the Voigt profile”, J. Quant. Spectrosc. Radiat. Transfer, 14, pp. 319-323, 1974], and the absorption due to the two line tails falling outside of the spectral band. The line tail calculations are performed for a line centered 0.2 cm−1 from the edge of the 1.0 cm−1 spectral bin; offsetting the location of the effective line from the center of the bin gives a more representative result for the average absorption of the two line tails. MODTRAN4 computes the line tail absorption by modeling the tail line-shape as being inversely proportional to the square of the line center displacement, i.e., ∝(Δv)−2. With lines centered 0.2 cm−1 from the edge of the spectral band, Doppler contributions to the line tails are small and the Lorentz line-shape denominator is dominated by the line center displacement term for Lorentz half-widths less than about 0.1 cm−1.
MODTRAN4 1.0 cm−1 band model line tail absorption is defined as the absorption from molecular transitions centered outside of the 1.0 cm−1 band but no more than 25 cm−1 from band center. Contributions from beyond 25 cm−1 are only considered for H2O and CO2, and modeled as continua (based on the approach in FASCODE). The line tail absorption is calculated from a database of temperature dependent 0.25 cm−1 integrated Lorentzian absorption coefficients. The line tail spectral dependence is assumed to be relatively flat so that the absorption coefficients can be modeled as constant over the 0.25 cm−1 spectral grid. To justify this assumption and enable line tails to be modeled as Lorentzian, atomic and molecular transitions centered too close to a 1.0 cm−1 spectral band edge are translated inward. A small line-shift correction is applied in-band to preserve the total integrated line strength. The overall error introduced into the 1.0 cm−1 band model by shifting line centers is small.
MODTRAN4 computes the total 1.0 cm−1 spectral band transmittance Tv for the neff identical lines by assuming line overlap characteristic of randomly distributed lines within a spectral interval. Plass [see: G. N. Plass, “Models for Spectral Band Absorption”, J. Opt. Soc. Am., 48, pp. 690-703, 1958] showed that the transmittance due to randomly distributed identical lines is given by the expression
                              T          v                =                                            (                              1                -                                                      W                    sl                                                        Δ                    ⁢                                                                                  ⁢                    v                                                              )                                      n              eff                                .                                    (        1        )            The Plass transmittance reduces to exact expressions in the limit of a single line, Tv(neff=1)=1−Wsl/Δv, and in the many line limit, Tv(neff→∞)=exp(−neffWsl/Δv).
As spectral resolution narrows, direct application of the MODTRAN4 band model becomes inaccurate. With a Δv=0.1 cm−1 spectral bandwidth, for example, and an effective average line positioned 0.25 Δv from the bin edge, line tail absorption at 1 atm pressure contains significant Voigt contributions, and the MODTRAN algorithm is not applicable in this regime.